We propose the following kinetic models to describe the evolution of ESR signals with temperature. A saturating system may be described by 1∂ñEa,t∂t=D̃1-ñEa,t-se-EakBT[ñEa,t], and a non-saturating system can be described by 2∂ñEa,t∂t=D̃ñEa,t-se-EakBT[ñEa,t], where 3ñt=∫0∞PEañEa,tdEa and 4PEa=1σ(Et)2πexp-12Ea-μ(Et)σ(Et)2. Here, ñ is the trapped-charge population with activation energy, Ea (eV). In the instance of a saturating system ñ is expressed as a saturation ratio, but for a non-saturating system it is expressed as absorbed radiation dose (Gy). The first term on the right-hand side of Eqs. () and () describes charge trapping as a 1st-order process. For a non-saturating system, D̃ is defined by the environmental dose rate D˙ (Gy), whereas for a saturating system, D̃ is defined as D˙/D0, where D0 is the characteristic dose of saturation (Gy). The second term on the right-hand side of Eqs. () and () describes thermal charge detrapping, and here we benefit from recent advances made in luminescence thermochronometry and follow Lambert (2018) by describing thermal detrapping using a model that assumes a Gaussian distribution of activation energies, σ(Et), around the mean trap depth, μEt(eV). Thermal detrapping is also described by the frequency factor s (s-1), the Boltzmann constant kB (eV), temperature T (K), and P(Ea) – the probability of thermally evicting electrons (or holes) from the trap (Eq. ). An alternative approach could be to use a 1st- or 2nd-order kinetic model as has been done previously (Toyoda and Ikeya, 1991; Ikeya, 1993; Grün et al., 1999), and we discuss our model selection more completely in the Supplement.

The natural trapped-charge concentration, which reflects the equilibrium between charge trapping and thermally stimulated charge detrapping, can be measured in the laboratory through the development of a sample-specific radiation dose response curve. This comprises measurement of a sample following increasingly large laboratory radiation doses and interpolation of the natural ESR signal onto the resultant dose response curve. Measurements can either be made on single aliquots (e.g. Tsukamoto et al., 2015) or using multiple aliquots of the same sample (e.g. Grün et al., 1999). The former approach has only recently been made practical, following the introduction of X-ray irradiation for regenerative dosing (Oppermann and Tsukamoto, 2015), as opposed to gamma irradiation, which is often done at a laboratory separate from the measurement laboratory.

Thermal detrapping can be measured following laboratory isothermal decay experiments, whereby aliquots of a sample are given a radiation dose before being heated at different temperatures for different durations. The resultant signal loss is measured and fitted with the kinetic model described in Eqs. (1)–(4). Previous investigations have suggested that the thermal decay of quartz ESR can be described by 1st-order or 2nd-order kinetics. Here, instead we use a density of states model, originally developed for the luminescence of feldspar (Li and Li, 2013; Lambert, 2018; further details of model selection are given in the Supplement). The selected model is based on a Gaussian distribution of activation energies σ(Et) around the mean trap depth, μ(Et) (Lambert, 2018), and may be applicable for quartz ESR data whereby electrons can be trapped in a variety of different defects, e.g. Ti3++e- charge compensated for by H+, Li+, or Na+ (Tsukamoto et al., 2018).

Five different monotonic cooling scenarios were used to test the potential of ESR thermochronometry in comparison to OSL thermochronometry, comprising cooling with rates of 100, 75, 50, and 25 ∘C Myr-1, as well as no cooling (i.e. isothermal holding at 0 ∘C for 2 Myr). All cooling rates were maintained for at least 2 Myr with a starting temperature of 200 ∘C, which is greater than the anticipated closure temperature of the ESR system (see Grün et al., 1999; Scherer et al., 1993, 1994). Using the kinetic model in Eqs. () and (), a trapped-charge population, ñfwd, was predicted for both the Ti and Al centres, respectively, using the kinetic parameters of sample KRG16-06 (Table 1) for the five different scenarios. In addition, the same exercise was carried out for four feldspar multi-OSL thermochronometry signals of the same sample using the following kinetic model after King et al. (2016a) (see the Supplement for further details on model selection): 5dñr′,Eb,tdt=D̃1-ñr′,Eb,t-se-Et-EbkBTñr′,Eb,t-s̃e-ρ′-13r′ñr′,Eb,t, where the total accumulation of charge with time, i.e. ñt, is obtained by integrating ñr′,Eb,t over the range of band-tail states, Eb, and an infinite range of dimensionless distances, r′: 6ñ(t)=∫r′=0∞∫Eb=0Etpr′PEbñr′,Eb,tdEbdr′, where P(Eb) is the probability of evicting electrons into band-tail states of energy Eb+dEb, defined as 7PEb=Be-EbEu, where B is a pre-exponential multiplier and where pr′ is the probability density distribution of the nearest recombination centre defined by Huntley (2006) as 8pr′dr=3r′2e-r′3dr′, where dimensionless distance r′≡4πρ313r, the dimensionless density of recombination centres ρ′≡4πρ3α3, and α is a constant related to the Bohr radius of the electron trap (Huntley, 2006; Kars et al., 2008; Tachiya and Mozumder, 1974).

We inverted the five sets of ñfwd values for the ESR and OSL data described above using a similar approach to King et al. (2016a), which we briefly outline here. The trapped-charge populations were modelled for 10 000 randomly generated time–temperature histories (t–T paths), which were constrained to cool monotonically between 200 ∘C and 0±5 ∘C over 2 Myr. We computed the dose response curves by solving the differential equations described above using a semi-implicit Euler method (Press, 2007). For each t-T path we calculated a misfit between the final inverted trapped-charge population, ñmod, and our forward-modelled values, ñfwd (Wheelock et al., 2015), from which the misfit M and likelihood L are calculated. 9M=Σ1m0.5ñfwdσlog⁡ñfwdñmod2 10L=exp(-M) This is for m traps, where σ is the uncertainty. An arbitrary uncertainty on ñfwd of 10 % was assumed. Cooling histories are then accepted or rejected by contrasting L with a random number between 0 and 1; if L is greater, the cooling history is retained. The accepted cooling histories are finally combined to construct a time–temperature history probability density function by dividing the time–temperature axis into 50 intervals and summing the number of paths that cross through each of the different cells. The Al, Ti, and OSL data were first inverted separately and then the Al and Ti centres were inverted together.

The results of the forward modelling and the synthetic inversions for the ESR and OSL data are shown in Fig. 1. The OSL signals for all cooling histories reach saturation (Fig. 1c), and this is reflected in the failure of the OSL to recover any of the cooling histories when inverted. This is apparent because the 1σ confidence intervals show a broad range, with the highest density of cooling histories concentrated at temperatures <20 ∘C over the past 500 kyr, indicating that the luminescence signals are saturated (as shown in Fig. 1c). The minimum cooling rate that can be resolved using OSL for sample KRG16-06 is ∼160 ∘C Myr-1, calculated from 86 % of the luminescence signal saturation level. Signal saturation is the key limitation that restricts the application of luminescence thermochronometry to regions undergoing rapid exhumation. In contrast, it is clear that the ESR data are able to resolve the 100, 75, and 50 ∘C Myr-1 synthetic cooling histories, and cooling rates of 25 ∘C Myr-1 are distinct from isothermal holding at 0 ∘C over timescales of ∼2 Myr. This is apparent because of the coincidence between the prescribed cooling histories (white lines) and the highest density of accepted cooling histories shown by the brightest colours in the probability density functions. These results are significant as they show that ESR thermochronometry is applicable in a range of geological settings beyond the rapidly exhuming locations that luminescence thermochronometry is currently restricted to.

Bedrock samples were prepared using standard laboratory methods under subdued red light conditions at the University of Lausanne and University of Bern, Switzerland (see King et al., 2016c). At least 10 mm was cut from the exterior of the samples using a water-cooled diamond saw to extract the light safe interior. A thin section was made using a representative sample of the bedrock exterior, and a further representative sample was sent to ActLabs, Canada, for ICP-MS analysis. Sample interiors were then hand crushed to extract the 180–212 µm grain size fraction, which was treated with HCl and H2O2 to remove any carbonates and organic material, respectively. The K-feldspar and quartz fractions were separated from heavy minerals using heavy liquids. The K-feldspars were retained for luminescence dating, whilst the quartz extracts (2.58>ρ<2.70 g cm-3) were etched for 40 min using 40 % HF, before being treated with HCl to remove fluorides that had precipitated during etching. The etched samples were sieved to >150 µm to remove any partially dissolved feldspar grains. Aliquots for ESR measurement comprised 60 mg of quartz loaded into glass tubes with interior and exterior diameters of 2 and 3 mm, respectively.

The grain size distribution of quartz and feldspar minerals within the parent bedrock was estimated from thin section analysis using the software of Buscombe (2013). The environmental dose rate, D˙, was calculated from the sample-specific radioisotope concentrations using DRAC v.1.2 (Durcan et al., 2015), the conversion factors of Guérin et al. (2011), the alpha grain size attenuation factors of Bell (1980), and the beta grain size attenuation factors of Guérin et al. (2012). Because the bedrock samples have only been at the surface for a short period of time, no cosmic dose rate was included in the calculation. The water content was estimated at 2%±2 %. For the quartz extract, an etch depth of 10 µm was assumed and the alpha dose rate adjusted following Bell (1980); an a value of 0.040±0.005 was used after Rees-Jones (1995) for any residual alpha dose. No internal dose rate was included. In contrast, the feldspar fraction was not etched, and an a value of 0.15±0.05 was used after Balescu and Lamothe (1994). An internal K content of 12.5%±5.0% was assumed following Huntley and Baril (1997). The calculated environmental dose rates are summarized in Table 1, and full calculation details are given in the Supplement.

Electron spin resonance measurements were done at the Leibniz Institute for Applied Geophysics in Hanover, Germany. Measurements were made on a JEOL JES-FA100 spectrometer using 2.0 mW microwave power, 0.1 mT modulation width, a 333.5±15 mT magnetic field, 0.1 s time constant, and 60 s scan which was averaged over three scans. All spectra were measured three times following sample turning by 60∘ to avoid any anisotropic effects. Measurements were made at -150 ∘C. The instrumentation detailed in Oppermann and Tsukamoto (2015) was used to facilitate X-ray irradiation and sample preheating, which is described below. The Ti and Al centre peaks were fitted using V3.3.35 of the JEOL ESR data processing software and were normalized relative to the intensity of the sixth hyperfine line of Mn2+ from the internal MgO standard, doped with MnO. As our measurements were carried out at -150 ∘C, it was not possible to differentiate between the Ti–H and Ti–Li centres, and consequently they have been treated as a single centre (Fig. 2). All subsequent data fitting was done using MATLAB.

OSL measurements of all samples followed the approach of King et al. (2016b, c). Luminescence measurements were made at the University of Bern using a single-aliquot regenerative dose multiple-elevated-temperature (MET) infrared stimulated luminescence (IRSL) measurement protocol (Li and Li, 2011) comprising a preheat at 250 ∘C for 60 s, followed by four IRSL measurements at 50, 100, 150, and 225 ∘C each of 100 s duration. A test dose of 160 Gy was used, which is ca. 30 % of the IRSL50 signal equivalent dose value of samples KRG16-06, KRG16-101, and KRG16-104. Each measurement cycle was followed by a high-temperature optical wash at 290 ∘C for 60 s. Regenerative doses up to ∼4.50 k Gy were given to three small (2 mm diameter) aliquots of each sample using two different Risø TL-DA-20 luminescence readers with dose rates ranging from 0.06 to 0.10 Gy s-1 dependent on the instrument (dose rates are provided for each measurement in the Supplement). Luminescence signals were detected in the blue part of the visible spectrum using a BG39 and BG3 or Corning 7-59 filter combination. The suitability of the selected measurement protocol was confirmed using a dose recovery test.

Rates of athermal and thermal charge detrapping were also measured using a single-aliquot regenerative dose method on the same aliquots used to measure the luminescence dose response curve. Athermal detrapping rates were quantified at room temperature by measuring the luminescence response to a fixed dose following different delay periods. Aliquots were preheated prior to storage following Auclair et al. (2003), and maximum fading delays were 122 d. Rates of thermal charge detrapping were measured using isothermal holding experiments. The aliquots were given a dose of 50 Gy and held at temperatures ranging from 170 to 350 ∘C for delay times of 0 to 10 240 s prior to measurement.

The signal intensity experiment indicates a plateau for the Ti centre of sample KRG16-104 up until 160 ∘C (Fig. 3a). In contrast, the Al centre for this sample and the Ti centre of sample KRG16-06 decrease in intensity by ∼5 % between room temperature and 160 ∘C, whilst the Al centre of sample KRG16-06 is depleted further, by ∼10 %. The signal intensity data for KRG16-06 are relatively noisy (non-monotonic signal decay with increasing preheat temperature) in comparison to KRG16-104. However, in spite of this, within the preheat plateau experiment (Fig. 3b), a plateau in De values between 160 and 220 ∘C is recorded for this sample following preheating for 2 min. On the basis of these experiments a preheat temperature of 160 ∘C was selected as this temperature maximizes signal intensity (Fig. 3a) whilst remaining within the De value plateau (Fig. 3b).

Preheating for short durations resulted in the heater unit overshooting the target temperature and poor thermal reproducibility. For this reason, a longer duration preheat at 160 ∘C for 4 min was selected for all measurements. This selected protocol is further validated by the successful recovery of a 360 Gy dose from naturally zero-age sample KRG16-112 for both the Al and Ti centres, which yield recovered-to-given dose ratios of 0.83±0.20 and 1.01±0.06, respectively (n=3).

Measurements of the trapped-charge population of the Al and Ti centres were similar between aliquots, resulting in 1σ uncertainties of ∼20 %. Equivalent dose values for the Al and Ti centres were within uncertainty for all samples, and ages ranged from 291±13ka for sample KRG16-05 to 36±3ka for sample KRG16-10 (Table 1). Samples KRG16-111 and KRG16-112 from the high-temperature tunnel yielded zero age; consequently, full dose response and isothermal decay was not measured for sample KRG16-112, and it is not included in Table 1. Whereas it was possible to saturate the Ti centre of all samples with the maximum given dose of 19 k Gy, the Al centre continued to grow linearly throughout measurement for all samples (Fig. 4a, c). Continued growth of the Al centre has been reported previously and has been accommodated through fitting dose response with an exponential plus linear function (e.g. Duval, 2012). In contrast, for the KRG samples, the Al centre is best described using a linear regression (Fig. 4a). The Ti centre of all samples showed a reduction in signal intensity at high doses (i.e. >10 k Gy), which has also been reported previously (e.g. Duval and Guilarte, 2015) and has been attributed to changing electron capture probabilities (Woda and Wagner, 2007). To characterize the maximum possible trapped-charge population we excluded data points at which the ESR signal intensity started to decrease (white data points in Fig. 4c) and fitted the remaining data with a single saturating exponential function (e.g. Grün and Rhodes, 1991) of the form 11ñ≈IIsat=(1-e-D+DeD0), where I is the natural ESR signal intensity, Isat is the saturation intensity of the ESR signal, and D is the given dose (Gy). Because the Ti centre experiences saturation, the equivalent dose value, De, can also be expressed as a saturation ratio, i.e. ñ=(n/N). As only a single aliquot of each sample was dosed until saturation, ñ values for the Ti centre were calculated from interpolation of the average De value (n=3) onto the single dose response curve; thus, ñ values for the Ti centre are derived from multiple aliquots.

Toyoda and Ikeya (1991) suggested that the thermal decay of the E', Al, and Ti centres follows 2nd-order kinetics; however, it was not possible to fit our data using either a 1st- or 2nd-order kinetic model (Supplement). Instead, the isothermal decay data were fitted using a multiple 1st-order kinetic model (Lambert, 2018; Table 1). Whilst the actual physical meaning of a Gaussian distribution of energies requires further investigation within the context of ESR defects, and it is unlikely that both the Al and Ti centres follow exactly the same process of thermal decay, preliminary fits to the data using this model are promising (Fig. 4b, d). Values of μ(Et) ranged from 1.27 to 1.90 eV between samples and centres (Table 1).

For all of the samples, the measured luminescence signals fulfilled the acceptance criteria (see the Supplement for further details). The IRSL50 signals of all samples exhibited very high rates of fading, with g2d values ranging from 6 % to 11 % decade-1, whereas for post-IR IRSL measurements at 225 ∘C, fading rates were 2 %–4 % decade-1. The model introduced by Huntley (2006) was used to fit the athermal detrapping data to determine ρ′. Using this model to fading-correct the trapped-charge concentrations following Kars et al. (2008) indicates that the IRSL50 signals of samples KRG16-06 and KRG16-101, and all signals for sample KRG16-05, are saturated (see the Supplement). All other signals can be used to determine rock cooling histories. Saturation of the IRSL50 signals relative to the higher-temperature signals is a consequence of their relatively high rate of anomalous fading. The luminescence dose response data for all of the samples and signals were fitted with a single saturating exponential fit to determine the characteristic dose of saturation, D0, and the concentration of trapped charge, ñ. Although for some samples a general-order kinetic (GOK) model fit would result in lower deviation from the measured values, GOK fits have been shown to overestimate sample athermal field saturation values (King et al., 2018), which must be done accurately to evaluate if a sample contains thermal information (see Valla et al., 2016). Finally, the isothermal decay data were fitted using the band-tail states model (Poolton et al., 2009; Li and Li, 2013; Eqs. 5–7) to determine Et, Eu, and s. Values of Et ranged from 1.27 to 1.52 eV between samples and signals (Table 2).

In order to invert the data into cooling histories, we used the same approach outlined in Sect. 3. We computed OSL and ESR dose response curves from 10 000 randomly generated t-T paths, which were constrained to cool monotonically between 200 ∘C and 15±5 ∘C over 2 Myr. Initially the Al centre, Ti centre, and OSL centres were inverted separately, before being inverted together. The results for all samples, with the exception of naturally zero-age samples KRG16-111 and KRG16-112, are shown in Fig. 5.